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On parameter differentiation for integral representations of associated Legendre functions. (English) Zbl 1218.31008

Summary: For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for associated Legendre functions of the first and second kind with respect to the degree are evaluated at odd-half-integer degrees, for general complex-orders, and derivatives with respect to the order are evaluated at integer-orders, for general complex-degrees. We also discuss the properties of the complex function \(f: C\{-1,1\}\rightarrow \mathbb C\) given by
\[ f(z)=\frac{z}{(z+1)^{1/2}(z - 1)^{1/2}}. \]

MSC:

31B10 Integral representations, integral operators, integral equations methods in higher dimensions
33B10 Exponential and trigonometric functions
33B15 Gamma, beta and polygamma functions
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)

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